Chapter 1 introduces pivotal concepts from functional analysis and control theory that form the basis for subsequent chapters. It covers essential topics such as cones and dual cones, semigroup theory, calculus of variations and optimal control, Pontryagin’s maximum principle, dynamic programming with viscosity solutions, and numerical methods for solving optimal control problems. By omitting less relevant results, the chapter ensures clarity and conciseness, providing readers with a solid mathematical foundation to deepen their understanding of advanced control theory and its applications.